Superconducting detectors for rare event searches in experimental astroparticle physics

TESs are among the most sensitive temperature sensors utilizing the properties of superconducting materials. A TES is a superconducting film operating at its superconducting-normal transition temperature ($T_\textrm{c}$). This superconducting film often consists of a single material (an elemental superconductor), such as tungsten (W). As an alternative, a bilayer consisting of a superconductor and a noble metal, such as Mo/Au, Mo/Cu, or Ti/Au, is also popularly chosen as the superconducting film. In the DM search experiments CDMS and CRESST, W-TESs have been used, while bilayer TESs are often chosen for high-resolution x-ray, gamma-ray, and alpha spectrometers. Using W or a bilayer as the superconducting film makes it possible to tune $T_\textrm{c}$. For example, the $T_\textrm{c}$ of a W film varies with its crystal structure (α-, β-, or γ-phase [148]) and the environment in which the film was deposited. The CRESST group uses α-W with $T_\textrm{c} \sim 15$ mK for their CaWO₄ crystals, and CDMS uses mixed-phase W with a $T_\textrm{c}$ of 50–100 mK in their detectors. In the case of a bilayer TES, its Tc can be adjusted via the proximity effect by varying the thickness ratio between the two layers [149].

The normal-state resistance of a TES is usually a few tens of mΩ, with a transition width of a few mK or less. Figure 8 shows a typical resistance curve of a TES as a function of temperature near the superconducting-normal phase transition 'edge'. The transition width can be on the order of 0.1 mK near a $T_\textrm{c}$ of approximately 100 mK. The temperature dependence of the resistance $(\partial R/\partial T)$ is very large at the transition, making the TES a very sensitive thermometer.

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TESs are, in general, operated in the voltage bias mode, such that a change in the resistance of the TES causes a change in the current in the bias circuit. This current change is measured with a low-noise SQUID with high accuracy. A simplified TES geometry and the corresponding measurement circuit are illustrated in figure 8. As shown in this figure, the TES film has a weak thermal connection to a heat bath that is regulated to a temperature below the $T_\textrm{c}$ of the film ($T_\textrm{bath} \lt T_\textrm{c}$). With the application of a bias voltage V that is sufficiently high to break the superconductivity of the TES film, the TES becomes resistive, and the temperature of the TES is elevated from $T_\textrm{bath}$ to within its superconducting-normal transition region due to Joule heating: ${P}_\textrm{Joule} = V^2/R(T)$, where R(T) is the resistance of the TES at T. With the weak thermal link being the only thermal connection between the film and the bath, the temperature of the TES is determined by the thermal differential equation:

$$\begin{equation} C \frac{dT}{dt} = - P_\textrm{bath} + P_\textrm{Joule} = - P_\textrm{bath} + \frac{V^2}{R(T)}, \end{equation} \tag{ 10 }$$

where $P_\textrm{bath}$ is the power flowing from the TES to the heat bath, which is a function of the TES temperature, the bath temperature, and the thermal conductance between the TES and the bath.

Equation (10) has important implications. When a TES is at a temperature T₀ within its transition region under a constant bias voltage and there is no external energy or power input, the TES is in thermal equilibrium, with $P_\textrm{bath} = P_\textrm{Joule} = V^2/R(T)$. However, once the temperature of the TES is increased (decreased) from T₀, the Joule heating decreases (increases) because R(T) has a positive slope in the transition region. This effect is called negative electrothermal feedback (ETF). As a result, the temperature of the TES is self-regulated at the quiescent temperature T₀. The negative ETF mode has several advantages for particle detection with TESs. Because they self-regulate their temperature within the transition region, a large array of TESs in a detector can be stably operated even if they have slightly different $T_\textrm{c}$ values, and they also become less sensitive to fluctuations in the bath temperature. Moreover, for a given energy input E, the recovery time of a TES becomes faster than its natural time constant in the absence of the ETF effect because the excess energy is effectively removed by a reduction in the Joule heating, thereby reducing the dead time of the detector due to pulse pile-up.

Another advantage of using a TES is that the TES film can be directly evaporated onto the surface of an absorber. Direct contact between the sensor and the absorber enables efficient heat transfer between them, which results in a much faster response time (a rise time of ∼1 ms) than those of other detectors such as thermistors (${\gt}10$ ms). The fast response time of TESs make them suitable for detecting athermal phonons which can help increase the detector sensitivity and facilitate the rejection of background signals.

TESs have been used as the temperature sensors in DM search projects such as CDMS and CRESST, as will be discussed in section 4.1. They have also been extensively developed for other applications, such as x-ray and nuclear spectroscopy. To the best of our knowledge, the best demonstrated FWHM energy resolutions are 0.7 eV at 1.5 keV [150] and 1.3–1.6 eV for 5.9 keV x-rays [143, 144], 65 eV for 208 keV gamma rays [151] and 1.0–1.1 keV for 5.3 MeV alpha particles [11, 152]. These energy resolutions are improved by orders of magnitudes compared to those of comparable commercial semiconductor detectors, such as Si(Li) detectors and high purity Ge detectors for x-rays and gamma rays and passive implanted planar Si detectors for alpha particles.

MMCs are the most mature magnetic thermometers used in particle detection. MMCs have become an important technology in various applications that require high energy resolution over a wide energy range [146], and they have great potential for use in rare event search experiments with crystal absorbers [153–155]. The best achieved energy resolution of an MMC to date is a 1.6 eV FWHM for 6 keV x-rays [145], which is comparable to the best resolution demonstrated by a TES. In an MMC setup designed for an energy range of 3–5 MeV, an FWHM of 0.86 keV has been obtained for the dominant Gaussian part of a 5.5 MeV alpha signal peak [12]. There is also another promising type of magnetic thermometer called a magnetic penetration thermometer (MPT) [156], but its maturity has not reached the level of MMCs, and it will not be discussed here.

An MMC uses a paramagnetic material as the temperature sensor. This is because the magnetization of a simple paramagnetic system is inversely proportional to the temperature, following Curie's law, and the temperature sensitivity can consequently be very large at low temperatures. For many years, Au:Er, a dilute magnetic alloy of gold doped with a small concentration of erbium, has been the most popular choice of the paramagnetic sensor material for MMCs because it maintains its paramagnetic properties at temperatures of tens of mK, and the good thermal conductivity of the host metal Au guarantees fast sensor thermalization at such low temperatures. Moreover, its thermal and magnetic properties are well understood through mean field theory, which takes into account the exchange interactions between the magnetic spins in a sensor [157].

However, there is a drawback in using Au:Er as the sensor material of an MMC because it shows excess heat capacity arising from nuclear quadrupole moments of the gold nuclei under the electrical field gradient induced by the Er ions [146, 157]. Since this excess heat capacity becomes the major contributor to the total heat capacity at temperatures below ∼20 mK, Au:Er is not an ideal sensor material for rare-event search experiments with large crystal absorbers typically running at 10–20 mK. As a promising alternative, Ag:Er, Er-doped silver [145, 158], has been quickly taking over the popularity of Au:Er. A recent study where low temperature properties of MMCs based on two types of sensor materials of Au:Er(1000 ppm) and Ag:Er(414 ppm) were measured [159] demonstrated the Ag:Er MMC showed no noticeable excess heat capacity and larger signal sizes than those from the Au:Er MMC.

MMCs utilize superconducting circuits and electronics to measure the changes in magnetization of the sensor material. Typically, superconducting coils are integrated on an MMC sensor chip to generate magnetic fields for magnetizing the sensor material and to pick up magnetization changes in the sensor. The signal picked up by the superconducting coils is read out by a dc-SQUID as a voltage signal. Thus, the detection principle of an MMC can be expressed as $E \rightarrow \Delta T \rightarrow \Delta M \rightarrow \Delta V$, where E is the input energy; $\Delta T$ and $\Delta M$ denote the change in sensor's temperature and magnetization, respectively; and $\Delta V$ is the voltage signal of the SQUID.

A simplified MMC setup is illustrated in figure 9. Initially, MMCs for particle detection were developed in a configuration in which a small piece of Au:Er attached to an absorber is placed inside the superconducting loop of a dc-SQUID. The SQUID loop itself was used as the pick-up coil of the MMC [146]. However, this early design imposed both fundamental and practical limitations. First, it was very challenging to efficiently couple the sensor and the pick-up coil in a reproducible way. It was also almost impossible to control the thermal conductance between the sensor and the heat bath and build a detector based on carefully worked-out thermal modeling. Furthermore, the magnetic field applied to magnetize the sensor could easily affect the performance of the SQUID. Most of these problems were overcome by lithographically fabricating a whole MMC sensor chip. This state-of-the-art MMC design includes an integrated superconducting pick-up/magnetization coil in the vicinity of the sensor layer, maximizing the coupling between them and allowing the use of ultralow-noise SQUIDs without their performance being affected by the magnetic field used for the operation of the MMC. The thermal link between the sensor and the heat bath is now controllable by on-chip thermalization pads.

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MMCs have many advantages for use as sensors in rare event search experiments. First, they have rather large heat capacities. Counterintuitively, this makes an MMC suitable as the sensor for an absorber with a large heat capacity because its performance will be less degraded by the added heat capacity of the absorber. Second, their working temperature range is very wide, permitting flexible optimization of the detector configuration depending on the detector performance requirements [155, 160]. MMCs also exhibit excellent detector linearity with a small deviation that can be reliably modeled.

The fast intrinsic thermalization of MMCs is another advantage for use in particle detection. The intrinsic thermalization time is affected by the interaction between conduction electrons and magnetic ions in the sensor material. When the absorber is strongly connected to the sensor layer or the sensor layer is used as its own absorber, an MMC can have an extremely fast signal rise-time. An MMC with a gold absorber fabricated on top of the sensor layer has demonstrated a rise-time shorter than 100 ns [161, 162].

In an application with a large crystal absorber, however, the rise-time of an MMC signal is much slower (on the order of ms) because it is primarily determined by the phonon generation mechanism and phonon dynamics in the crystal as well as the thermal connection between the absorber and sensor rather than by the intrinsic thermalization of the sensor [153]. Nevertheless, even with the slow ms-scale overall rise-time with a scintillating crystal absorber, MMC signals are sufficiently fast to distinguish the pulse shape difference between electron- and alpha-induced events, which are considered signal and background events, respectively, in $0\nu\beta\beta$ experiments [155]. Moreover, a rise-time on this scale, which is approximately 1–2 orders of magnitude faster than that of thermistor-based detectors, can help reduce the unresolved pile-up (random coincidence) event rate, which can be one of the most significant background sources for slow detectors in $0\nu\beta\beta$ experiments [163].

A KID is a superconducting sensor with a very different detection mechanism from those of the detectors discussed above. It is considered a nonequilibrium detector and uses a change in kinetic inductance in a superconductor as its main signal.

Superconductors enter the resistance-free state under a dc current (smaller than a certain critical current) below a certain transition temperature. However, they exhibit a nonzero resistance under an ac current due to a complex surface impedance (or kinetic inductance) phenomenon. When a superconducting strip, cooled below its $T_\textrm{c}$, absorbs energy from a particle or radiation, some of the Cooper pairs in the strip are broken, generating quasiparticles. The higher the input energy is, the more quasiparticles will be generated. These excess quasiparticles change the kinetic inductance of a microwave resonance circuit, which can be measured as a shift in the amplitude and phase of the resonance, as shown in figure 10 [164]. This is the basic working principle of a KID. For the superconducting strip materials in these detectors, Al, Al/Ti/Al, Hf, TiN, Ti/TiN, and PtSi are often used [165, 166].

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The greatest advantage of KIDs is that they enable straightforward multiplexing of many channels. A typical KID consists of many resonance circuits with a high quality factor. By spreading the resonance frequency of each resonance circuit over a large bandwidth (e.g. 2 MHz spacing over a 4 GHz bandwidth), it becomes possible to send a comb of frequency tones corresponding to each resonator into the detector via a single feedline, and its response can then be measured by room-temperature electronics. In principle, more than a thousand resonators can be read out by a single feedline. Because of their highly multiplexing capability and low noise, KIDs have already been employed in many instruments to detect astronomical signals ranging from the millimeter scale to the ultraviolet (UV) [167, 168].

In rare event search experiments, KIDs have not received as much attention as in astronomical applications, mainly because when an KID is integrated as a phonon detector (a thermal KID, or TKID), its energy resolution in the x-ray regime is not sufficient to be competitive with other devices such as TESs [169]. However, as the focus of DM searches moves towards lowering the detection threshold, KIDs have arisen as an attractive solution. When a coplanar waveguide feedline with many KID resonators is fabricated on a side of an absorber wafer, the initial athermal phonons may break Cooper pairs in the resonators near the event location. Each resonator would behaves like a low- threshold phonon sensor on a target absorber helping position reconstruction of the rare events [170] as in the R&D study of the BULky and Low-threshold Kinetic Inductance Detectors (BULLKID) project [171]. Moreover, in the Cryogenic wide-Area Light Detectors with Excellent Resolution (CALDER) project [172], KIDs are also being developed as low-energy photon detectors for a $0\nu\beta\beta$ experiment.

There are a few LTDs that do not use superconducting materials or superconducting circuits but are popularly used in rare event search experiments. One such detector is a doped semiconductor thermistor with a strongly temperature-dependent resistance. Neutron-transmutation-doped (NTD) Ge thermistors and ion-implanted Si thermistors are the two most common types of thermistors used at low temperatures.

When a pure semiconductor such as Si or Ge is doped at a concentration slightly below the critical doping for the metal–insulator transition, its low-temperature resistance is governed by a process called Efros-Shklovskii (ES) variable-range hopping [173] and can be modeled as:

$$\begin{equation} R(T) = R_0 \exp \left( \frac{T_0}{T} \right)^{1/2}, \end{equation} \tag{ 11 }$$

with two characteristic constants, R₀ and T₀ [174]. The resistance of a typical thermistor is 1–100 MΩ near 50 mK.

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Such thermistors are fairly easy to use because they can be operated with conventional electronics such as junction field effect transistors (JFETs) rather than requiring sophisticated superconducting electronics. They are typically current biased, and the measured voltage across such a thermistor is amplified by a JFET located at ∼100 K or at room temperature.

Of the two types of thermistors, NTD Ge thermistors are preferred in large-scale experiments primarily because of their reproducibility and uniformity. For example, the Cryogenic Underground Observatory for Rare Events (CUORE) collaboration uses NTD Ge thermistors as the phonon sensors in their kilo-pixel ton-scale detector. On the other hand, ion-implanted Si thermistors are preferred in experiments where the individual detectors are small in size with delicate electrical and mechanical structures because the dopant ions can be implanted in a desired pattern through masking, and techniques for the micromachining of silicon are well established. Thus, Si thermistors have been a popular choice for spaceborne x-ray missions [174].

Both Ge- and Si- based thermistors have shown reasonable energy resolutions in different energy regions. For instance, NTD Ge thermistors have achieved a FWHM resolution of 3.1 eV at 5.9 keV x-rays [175], and Si thermistors have achieved resolutions of 3.2 eV at 5.9 and 22 eV at 60 keV [176] (compared to the best TES resolution of 1.3–1.6 eV and the best MMC resolution of 1.6 eV at 6 keV and 10.2 eV at 42 keV [177]).

One drawback of thermistors is their rather slow response time, which is caused by poor coupling between the conduction electrons and the phonons in the thermistor. This slow response is unfavorable in experiments where the rate of random coincidence events should be minimized as in $0\nu\beta\beta$ search experiments.

The heat transfer between two different media was studied theoretically and experimentally in the 1960s–1980s [178]. In particular, the heat transfer associated with thermal contact resistance, known as the Kapitza resistance (or the inverse of the Kapitza conductance), was investigated in many experiments involving liquid helium, various metals, and dielectrics. One of the most successful models for the contact resistance was found to be the acoustic mismatch model suggested by Little [179]. In this model, a partial transmission of phonons across an interface between two media with different densities and sound velocities is regarded as the origin of the Kapitza resistance at low temperatures. In other words, the thermal resistance between the two media is considered to be caused by the difference in their acoustic impedances, i.e. the acoustic mismatch.

The essential result from the acoustic mismatch model is that the thermal conductance $G_\textrm{K}$ of the heat transfer is proportional to the contact area A of the interface and has a cubic temperature (T) dependence, expressed as:

$$\begin{equation} G_\textrm{K} = \Lambda A T^3, \end{equation} \tag{ 12 }$$

where Λ is a proportionality constant that can be calculated based on the model [178, 180]. A number of experiments involving interfaces between various media show reasonable agreement with the predictions of this model [178, 181].

Studies of the interactions between electrons and phonons in metals have also been a popular subject in modern physics. For instance, electron–phonon interactions play a key role in explaining the mechanism of superconductivity, which arises from indirect electron–electron interactions mediated by phonons. Another significant implication of electron–phonon interactions is that they change the effective mass of electrons in metals, offering explanations for many characteristic properties of metals. Furthermore, electron-phonon interactions are related to ultrasonic attenuation and the hot electron effect in metals.

In the context of modeling the heat flow in metals at low temperatures, we present a brief introduction to the hot electron effect originating from electron-phonon relaxation processes in metals. The hot electron effect, a temperature decoupling between the electrons and the lattice in a metal, is a phenomenon that can be observed not only at low temperatures but also at much higher temperatures. Many experiments using short laser pulses incident on metal films have demonstrated the hot electron effect. For example, Schoenlein et al demonstrated that a high-intensity femtosecond laser pulse could induce hot electrons at a temperature higher than 1000 K in gold, while the lattice temperature remained at approximately 300 K [182]. Allen constructed a theory for calculating the temperature relaxation of such hot electrons via electron-phonon collisions governed by the Bloch–Boltzmann–Peierls formulas [183]. The calculation assumes that the electrons first thermalize among themselves as a result of heating from the short laser pulse and then subsequently lose their energy to the lattice. An important conclusion of this theory is that the heat transfer from the hot electrons to the lattice is proportional to ($T\,_\textrm{e}^5-T\,_\textrm{l}^5$), where $T_\textrm{e}$ and $T_\textrm{l}$ are the temperatures of the electrons and the lattice of the metal, respectively. Allen claimed that this theory is also applicable at low temperatures.

Accordingly, the heat transfer through the electron-phonon coupling in a metal can be expressed as:

$$\begin{equation} P_\textrm{ep} = V \Sigma (T\,_\textrm{e}^5- T\,_\textrm{l}^5), \end{equation} \tag{ 13 }$$

where V is the volume of the metal and Σ is a material dependent parameter that incorporates a coupling constant for the electron-phonon interactions in the metal. According to Allen's theory, Σ is approximately $2\times10^9$ W m⁻³ K⁻⁵ for bulk gold based on a coupling constant found from high temperature measurements [183]. For a small temperature difference between T_e and T_l , the corresponding thermal conductance of the electron-phonon coupling can be approximated as:

$$\begin{equation} G_\textrm{ep} = 5 V \Sigma T_e^4. \end{equation} \tag{ 14 }$$

Based on the above discussion, the heat flow between a metal film and a dielectric substrate primarily consists of two heat flow mechanisms connected in series: the Kapitza resistance from the acoustic mismatch of the materials and the impedance due to electron–phonon interactions. The conductance arising from the series combination of these two thermal impedances is:

$$\begin{equation} G = \left( \frac{1}{G_\textrm{K}} + \frac{1}{G_\textrm{ep}} \right)^{-1}. \end{equation} \tag{ 15 }$$

Since $G_\textrm{K}$ is proportional to AT³, and $G_\textrm{ep}$ is proportional to VT⁴, $G_\textrm{K}$ governs the conductance between the metal film and the substrate at high temperatures. However, at sufficiently low temperatures, G is limited by $G_\textrm{ep}$. When designing a detector with a metal film, it is often useful to know at which temperature the two conductances become equal, i.e. $G_\textrm{K} \approx G_\textrm{ep}$. For a metal film with a thickness of $t = V/A$, the equal-conductance temperature T in K is approximately the inverse of t in µm.

The heat flow mechanism along a metal film or other metal structure depends on the thermal conductivity of the conduction electrons. According to diffusion theory for an electron gas, the electronic thermal conductivity κ_e is equal to $\frac{1}{3} c v_\mathrm{F} \lambda$, where c is the specific heat, $v_\mathrm{F}$ is the Fermi velocity, and λ is the mean free path of the electrons. At sufficiently low temperatures, because the electron mean free path is temperature independent, the thermal conductivity is proportional to the specific heat of the metal. This results in the electronic thermal conductivity being proportional to the temperature. Moreover, the electronic thermal conductivity has a simple relation to the electrical conductivity σ for most noble metals at low temperatures, as follows:

$$\begin{equation} \frac{\kappa_e} {\sigma} = L_0 T, \end{equation} \tag{ 16 }$$

where L₀ is the Lorentz number, which is equal to 2.45 × 10⁻⁸ WΩ K⁻². This relation is known as the Wiedemann–Franz law. It indicates that the electronic thermal conductance of a metallic structure can be estimated from its electrical resistance and geometry. In general, the electrical resistivity ρ of a metal decreases with decreasing temperature until, at very low temperatures, it reaches a finite value due to the small amounts of impurities and defects in the metal. The residual resistivity, $\rho_\textrm{4 K}$, also depends on the microstructure of the metal such as the grain size. When fabricating metal films for LTD applications, the residual resistivity ratio (i.e. $\rho_\textrm{300 K}/\rho_\textrm{4 K}$) is commonly used as an indicator of film quality and is often maximized to achieve high thermal conductivity at low temperatures.

The CDMS is an international collaboration for direct detection of WIMPs that has pioneered many important superconducting detector technologies. In particular, it is well known for its major contributions to the development of TES and heat-ionization dual-channel detectors based on quasiparticle diffusion in thin films and the NTL effect in semiconductor absorbers. Most of these techniques were already discussed in the previous section, but we will briefly revisit them when applicable.

The first phase of CDMS was called CDMS I (1998–2002), which was followed by CDMS II (2003–2009) and then SuperCDMS Soudan (2011–2015). The current phase is SuperCDMS SNOLAB, and a detector is being built in the deep underground laboratory at SNOLAB.

With each phase of the experiment, the number and the total mass of the target crystals increased. CDMS I included 6 detector modules composed of a Ge crystal, with a total mass of 1 kg. The experiment was conducted at a shallow underground laboratory (SUF, ${\sim}10$ mwe (meter water equivalent)) and achieved a total exposure of 30 kgd. CDMS II included 30 detector modules (4 kg of Ge in total) in a deeper underground laboratory of ∼2100 mwe in Soudan. With 400 kgd of exposure, CDMS II resulted in the then strongest limit on the WIMP-nucleon spin-independent scattering cross section, nearly rejecting the DAMA/LIBRA annual modulation region. SuperCDMS Soudan included 15 detector modules with a total Ge mass of 9 kg and was also conducted in Soudan. The next-generation experiment SuperCDMS SNOLAB will be operated on the order of 100 detector modules totaling approximately 200 kg. This latest phase aims to probe not only the traditional WIMP mass region but also the light DM region. For the latter, a fraction of the detectors are configured to have an ultralow detection threshold.

With each successive phase, the CDMS detector design also continued to evolve. However, the core measurement scheme remained the same: the detector consists of multiple detector modules operating at cryogenic temperatures, each module is equipped with a sizeable disk-shaped Ge or Si crystal as the target for DM-normal matter interactions, and the temperature and ionization signals induced by an energy input in the target crystal are simultaneously measured by a temperature sensor and a field effect transistor (FET) amplifier, respectively, as illustrated in figure 13. The dual-channel measurement makes it possible to distinguish NR signals from electron recoil signals, thus significantly improving the detector sensitivity to NR-like DM signals, as previously discussed (see figure 14).

The temperature sensors in the CDMS detectors, except for that in one early generation called the Berkeley Large Ionization- and Phonon-mediated (BLIP) detector, which is based on NTD Ge sensors, consist of thousands of tiny W TESs spread over the surface of the target crystal in a hierarchical arrangement, a unique feature of CDMS's Z-sensitive ionization and phonon-mediated (ZIP) detector. More specifically, CDMS I ZIP contained 4 independent phonon channels, with each channel consisting of 37 cells and each cell consisting of 12 TES elements coupled to Al 'fins' or 12 quasiparticle-assisted ETF transition-edge sensors (QETs) as shown in figure 15(a). In total, there were $4\times444$ TES elements for each target crystal. The TESs had a $T_\textrm{c}$ of ${\sim}100$ mK and were voltage-biased such that they were in the extreme ETF mode. This ensured stable operation of such a large number of TES elements. The Al fins served as quasiparticle collectors as well as the ground electrode of the ionization detector and covered 82% of the top surface of the crystal. The phonon signal was read out by SQUID arrays.

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The CDMS II ZIP detector had a layout very similar to that of the CDMS I ZIP detector but with an improved phonon collection efficiency. The QETs of the CDMS I ZIP detector had a fill factor of nearly 100% near the QET coverage area (82% of the top surface); because the average distance between generated quasiparticles and the nearest TES was larger than the diffusion length of the quasiparticles, some quasiparticles were unable to reach a TES within the time constant of the TES. To solve this problem, the number of TESs per channel was increased from 12 to 28, and the length of the Al fins was decreased in the CDMS II ZIP detector as shown in figure 15(b). There were even more upgrades in the SuperCDMS detectors, as described below.

The ionization channel of the CDMS detectors consists of charge-collection electrodes deposited on both sides of the target crystal, which are used to generate the electric field needed for the created e–h pairs to drift along the field lines and to be collected in the electrodes [201]. To reduce the loss of the ionization signal for surface events, a thin layer of amorphous silicon was deposited between the electrode and the target crystal in every detector except CDMS I ZIP [199]. In the CDMS I and II detectors, the electrode on the top surface (the 'phonon side') served as the ground electrode, and the electrode on the bottom surface (the 'ionization side') was biased and connected to a current integrator. The generated e–h pairs under electric fields created NTL phonons, which contributed to the total phonon signal observed by the phonon sensor, as illustrated in section 3.6. To prevent a gradual drop in gain of the NTL phonon amplification due to charged trap centers in the crystals, LEDs were used to routinely neutralize the crystals. Moreover, to avoid creating too many NTL phonons, which would compromise the detector's PID capability, a relatively small bias voltage of a few V was used in CDMS I and II.

In addition to exhibiting PID capability, the CDMS detectors have position sensitivity, which can be utilized to further reject the background signal. As described above, the CDMS I and II ZIP detectors have phonon sensors segmented into four individual channels. By analyzing the pulse shape of a phonon signal from the four channels (e.g. by comparing the arrival times), the lateral position of the signal can be determined. Furthermore, the pulse-shape difference between the bulk and surface events can be used to reject the background signal occurring in the detector's 'dead layer' near the surface, where the sensitivity of the ionization channel is compromised.

Now, we describe the SuperCDMS detectors. Based on the basic working principle of the CDMS I and II ZIP detectors, SuperCDMS has involved significant upgrades in its new interleaved Z-sensitive Ionization and Phonon sensors (iZIP) detector. In this new detector, phonon sensors are patterned on both sides of the crystal and interleaved with ionization electrode lines, as shown in figure 16. The phonon sensors on the top and bottom surfaces are separated into four channels in the SuperCDMS Soudan iZIP and six channels in the SuperCDMS SNOLAB iZIP, covering the entire top and bottom surfaces of the crystal. The phonon sensors also serve as the ground electrodes in both ZIPs and iZIPs. However, unlike ZIP detectors, iZIP detectors have both positive (+2 V) and negative (−2 V) bias electrodes. These changes were made to improve the sensitivity of the iZIP detectors to surface events and to better extract the energy and position information of each event.

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There are several variants of the SuperCDMS Soudan and SNOLAB iZIP detectors, such as CDMSlite and CDMS-HV. The main characteristic of these variants is that they use a much higher bias voltage (${\gt}$50 V) rather than a few V. This higher voltage increases the gain of NTL amplification, thus improving the energy resolution of the phonon channel and decreasing the energy threshold at the expense of compromised PID capability. This was intended to enable probing of the low mass region of the DM parameter space. SuperCDMS-CPD, a new detector developed in collaboration with the Cryogenic PhotoDetector (CPD) project [203], has taken a different approach. SuperCDMS-CPD uses a small (approximately 10 g) Si crystal rather than massive Ge crystals. The use of such a small crystal resulted in a substantially improved baseline energy resolution of approximately 3.9 eV [68], which resulted in a promising limit curve, as shown in figure 4.

The Cryogenic Rare Event Search with Superconducting Thermometers (CRESST) is another multi-institution direct DM search experiment that uses TESs as sensors of the DM-normal matter interaction. CRESST is known for its use of scintillating crystals equipped with phonon-photon channel detection, as shown in figure 13. The collaboration recently completed the operation of CRESST-III Phase 1 [204] and is in the preparation stage for CRESST-III Phase 2.

The first CRESST experiment (CRESST-I) used four sapphire (Al₂O₃) crystals (262 g each) as the target material for the detection of DM-nucleon scattering events [205]. Sapphire crystals were chosen because of their high Debye temperature [206], which can make the specific heat of the detector very small at cryogenic temperatures. To increase the resolution and sensitivity of the detector, the operating temperature was decreased to as low as ∼15 mK. To match this low temperature, a TES composed of a W film (or a superconducting phase transition thermometer, as it was called within the collaboration [207]) was developed. To achieve this very low $T_\mathrm{c}$, an α-phase W film was electron beam evaporated on heated sapphire crystals in an ultrahigh vacuum environment [148]. For stable operation of the detector at the $T_\mathrm{c}$ of the W film of ∼15 mK, an active thermal feedback scheme in which the base temperature of the cryostat was maintained at ∼6.5 mK and an additional heater integrated into the detector maintained the detector temperature at the $T_\mathrm{c}$ [208] was adopted, as shown in figure 17. The heater also served as a means to calibrate the detector.

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The first DM limit result from the CRESST-I detector was released in 2002 [209]. Although these detectors had the highest sensitivity per unit mass among cryogenic detectors in use at the time, this result did not have sufficient sensitivity to rule out the 'DAMA' region of WIMP's annual modulation. This limitation was mainly ascribed to the inability to discriminate NR-like signals from gamma-dominated background events. The collaboration implemented a PID solution to increase the detection sensitivity by employing simultaneous detection of heat (phonon) and light (photon) signals [210], which is a slightly different scheme from that of the CDMS, whose PID mechanism is based on simultaneous detection of heat (phonon) and ionization (charge) signals.

In certain scintillating crystals, electron- and photon-induced signals show a different light yield than that due to heavy ions or neutrons, as shown in figure 14. Among several different scintillating crystals the CRESST collaboration tested, CaWO₄ showed the best performance at cryogenic temperatures. Based on this finding, they made significant changes in their detector design (CRESST-II): the sapphire crystals were replaced with CaWO₄, and light detectors were added, as shown in figure 18.

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Note that the layout of the light detector is slightly different from that of the phonon detector. As shown on the right side of the figure, Al phonon collectors with large areas, which were meant to maximize the signal size for a given energy deposition, were added.

The results of the commissioning phase of CRESST-II (completed in 2007) as well as those of the subsequent CRESST-II phase 1 (2009–2011) showed improved DM limits that excluded the DAMA region [211–213]. This result was consistent with the null finding of other DM detectors, including the two LTD experiments of CDMS and EDELWEISS. The design of the detector module was updated in four different ways in CRESST-II phase 2 to clarify some events with no or small amplitudes in the light channels that were unlikely from DM scatterings or leakage events due to known backgrounds. One 300 g module of CRESST-II phase 2 showed a 300 eV threshold, resulting in a large improvement in the DM detection sensitivity [214, 215]. However, CRESST-II observed excess events in the 'acceptance' region in both experimental phases, which were interpreted as background instead of positive WIMP signals.

The most recently completed CRESST-III phase 1 (2016–2018) focused on the low-mass WIMP region. For this purpose, the mass of the CaWO₄ crystals was reduced to 24 g each (a factor of 10 reduction), significantly lowering the detector threshold to below 50 eV [216]. Moreover, the new modules were designed to employ crystal supporting rods composed of CaWO₄, and each rod had a TES film, as shown in figure 19. This was to veto any events caused by the thermal signal propagating to the target crystal through its holding structure, which could explain the excess events in the acceptance region.

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CRESST-III phase 1, with its lower threshold, resulted in an improvement in the DM limit in the low-mass region (∼500 MeV c⁻² and below) by an order of magnitude. A new run of CRESST-III phase 2 with considerably increased exposure is planned, which is expected to further extend the low-mass region of the DM parameter space.

Based on the sensor technology developed for CRESST, another DM search project, COSINUS, has been proposed to cross validate the 14 year long annual modulation results of the DAMA/LIBRA [217, 218]. This project is aiming to develop a heat-light detector setup with a detection threshold as low as 1 keV using CRESST-style TES sensors and NaI scintillating crystals, a type of crystals that DAMA/Libra measured with conventional photo-multiplier tubes [59].

Using TESs, CDMS and CRESST have shown remarkable progress and promising results in direct DM detection covering the low-mass region of the DM parameter space. Expérience pour DEtecter Les WIMPs En Site Souterrain (EDELWEISS), another DM search project based on LTDs, utilizes NTD Ge thermistors coupled to Ge target crystals as the phonon sensor. The Ge crystals are also equipped with electrodes to measure the ionization signal. This two-channel detection allows distinction of NR events from electron recoils in the target crystal.

EDELWEISS included three measurement phases, all of which were carried out at the Modane Underground Laboratory in France. EDELWEISS-I, the first phase, consisted of three detector modules composed of 320 g Ge crystals (approximately 1 kg in total). EDELWEISS-I resulted in the first rejection of DAMA's positive claim of DM detection [219]. After this result, many other experiments, including CDMS and CRESST, also ruled out DAMA's claim. In EDELWEISS-II, the collaboration operated 10 detector modules of 400 g Ge crystals with a sensitivity of $4.4\times10^{-32}$ cm² for a DM mass of 85 GeV [220]. In EDELWEISS-III, 24 detector modules of 820–890 g Ge crystals were operated to search for DM in the mass range of 1–20 GeV [221]. All three phases yielded null results of the DM signal. Currently, as with CDMS and CRESST, the EDELWEISS collaboration is shifting its focus to the detection of low-mass DM [221].

EDELWEISS is acclaimed for the development of an efficient method to reject surface events, which are one of major limiting factors of the sensitivity of DM detection in the phonon channel. Surface events on a Si or Ge target crystal may undergo incomplete charge collection, which can result in misidentification of electron-induced surface events by NR events because, although due to a different mechanism, NRs also give rise to a relatively small ionization signal for a given energy input. To solve this issue, EDELWEISS designed a new detector with two sets of interdigitated electrodes patterned on the surface of the Ge crystal [222, 223]. One set of electrodes is used to collect ionization signals from bulk events, and the other set is used to veto surface events. In the latest version of the detector, called the fully interdigitated detector (FID), the electrodes cover every surface of a crystal to improve the event discrimination efficiency. The FID has demonstrated more than 99% efficiency [41]. Similar interdigitating electrodes were also adopted in SuperCDMS's iZIP detectors, as discussed in the previous section.

During recent decades, significant progress has been made in the development and understanding of low-dimensional devices [224]. Many novel low-dimensional devices are based on superconducting technologies and operate at sub-Kelvin temperatures. These devices were motivated by various practical applications, such as quantum computing and memory devices, but have also found their way into astroparticle physics as detectors of rare event search experiments. Here, we briefly introduce detectors based on low-dimensional devices (low-D detectors).

In thermal calorimetric detection, a smaller heat capacity increases the signal size, which is set by the temperature change for a given input energy, as described in equation (6). One way to maximize the temperature increase is to make a detector without an absorber and use the sensor itself as an absorber. The heat capacity of such detectors can be further reduced by shrinking the dimensions of the sensors. Thus, low-D nanometer-scale detectors have become a promising choice for calorimetric detection of DM.

These nanometer-scale thermal detectors, however, are not always a viable solution for the direct detection of DM, especially for WIMP-like DM particles with a mass of approximately 100 GeV, because the small overall size of the detector prevents it from being competitive in this mass region. On the other hand, they can be an excellent choice for light DM because of their extremely low detection threshold. Low-D detectors based on tunnel junctions are good examples.

The Josephson-threshold calorimeter (JTC) is a low-temperature thermal detector based on a temperature-biased tunnel Josephson junction formed by different superconductors [225]. A Josephson junction is a device consisting of two or more superconductors connected by a weak link typically made of an insulator or a normal metal that operates based on the Josephson effect [226]. Josephson junctions have tremendous applications, especially in quantum sensing. For JTCs, Josephson junctions engineered with two superconductors of different $T_\textrm{c}$'s and different superconducting gap Δ's are employed to have a sharp critical current dependence on the temperature. Using the lower $T_\textrm{c}$ junction element as the sensor/absorber, JTCs are expected to be capable of photon number resolution for photons from the mid-IR region through ultraviolet region (frequencies in the range of 30 THz to $9\times10^4$ THz).

The graphene Josephson junction (GJJ) detector is another type of low-D detector suitable for light DM. A GJJ uses a graphene film as the sensing element [227, 228]. It utilizes the peculiar property of graphene in which the electron density of states vanishes at the charge-neutrality point, which leads to a highly suppressed electronic specific heat [229]. Recently, a DM experiment was proposed based on GJJ detectors that can probe the DM parameter space down to 0.1 keV [230].

The quantum capacitance detector is another low-temperature detector capable of low-energy photon counting. When a photon is absorbed in a mesh absorber composed of superconducting aluminum, the photon produces free electrons, which tunnel into a single Cooper-pair box (SCB) island, a variable capacitor embedded in a resonant circuit. A change in the SCB state due to electron tunneling is recorded an RF signal. Recently, THz single photon detection was demonstrated using this technique [231].

These small low-temperature devices are mostly in development and have not yet been deployed in an active DM search experiment. However, as an increasing number of DM search experiments report null results in the traditional WIMP-like DM mass region and the focus shifts toward lower mass DM, these detectors are expected to play a crucial role in next-generation DM search experiments.

The Advanced Mo-based Rare-event search Experiment (AMoRE) is a large-scale international project to search for $0\nu\beta\beta$ of ¹⁰⁰Mo in a deep underground laboratory. Similar to the previously discussed DM detection experiments, AMoRE employs superconducting detectors to probe extremely rare $0\nu\beta\beta$ signals. The AMoRE detectors consist of heat and light detectors to measure the heat and light signals, respectively, induced by energy input in scintillating crystals, a detection scheme capable of PID analogous to that of CRESST. However, unlike DM detectors, AMoRE detectors are optimized to measure electron-induced events on the MeV energy scale.

AMoRE takes advantage of the high Q-value of $2\nu\beta\beta$ of ¹⁰⁰Mo (3034 keV). Because this Q-value is greater than the energy of most environmental γ-rays, especially the ²⁰⁸Tl line of 2615 keV, the background rate in the region of interest (ROI) is significantly reduced. Nevertheless, another source of background due to α decays inside or near the absorber crystal may overwhelm the possible $0\nu\beta\beta$ signal unless it is properly eliminated. This is why the AMoRE detectors require PID capability for background rejection in addition to a high energy resolution.

AMoRE has involved tremendous effort in the development of heat and light detectors as well as their target absorber crystals. The heat and light detectors of AMoRE both employ high-resolution MMC sensors. For the absorber, among several candidates, CaMoO₄ and Li₂MoO₄ have been chosen based on their high light yield and reasonable cost for mass production. In the first cryogenic test of a small CaMoO₄ with MMC, reasonable energy resolution in a wide energy range was achieved, demonstrating the applicability of the CaMoO₄–MMC combination detector [154]. Soon, a detector with much larger crystals of CaMoO₄ and Li₂MoO₄ (${\gt}200$ g) equipped with a separate light detector was successfully built, which became the basis of the AMoRE detector module [234, 235].

The AMoRE detector consists of multiple modules, each consisting of a target crystal of CaMoO₄ or Li₂MoO₄ and a light detector enclosed by highly light-reflecting material, as shown in figure 20. In each module, two MMC sensors are used, one for the heat (phonon) channel and the other for the light channel. To increase the sensitivity of the heat channel, which is AMoRE's main signal, a gold film that serves as a phonon collector is evaporated on one side of an absorber crystal. Then, an MMC sensor is thermally connected to the phonon collector film via gold bonding wires. As discussed in the section 3, the input energy in the crystal is transferred to the MMC sensor via athermal and thermal processes of heat flow. The athermal phonons generated in the crystal in the very early stage of energy absorption travel around the crystal and have a significant probability of directly depositing their energy onto the phonon collector before downconversion to thermal phonons. The heat transferred to the phonon collector flows into the MMC sensor and eventually to the heat bath. The light detector is built on the basis of an almost identical heat flow mechanism, with two main differences: a Si or Ge wafer is used as the absorber for lights from the main scintillating crystal, and input energy, deposited by thousands of photons, is absorbed in a broad area than being localized, as is the case for the heat detector.

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The AMoRE detectors were carefully designed based on complete thermal modeling that involves Kapitza conductance, electron-phonon interaction, and electronic heat flow following the Wiedenmann–Franz law [189]. Thus, optimized AMoRE detectors have shown an energy resolution of ∼15 keV FWHM at 3 MeV. Although 15 keV FWHM is considered a high resolution for a large detector, it can be further improved by reducing the vibration of the cryostat [236] and by reducing the position dependency [237]. In the current design, due to the finite thermalization time of the large crystal, the signal size for a given energy input varies depending on the position of the input, thus decreasing the energy resolution. Employing an additional MMC sensor is expected to not only solve the position dependency problem but also provide extra information on the position of the event through the signal size difference between the two sensors. This extra information can be used to distinguish localized $0\nu\beta\beta$ events from delocalized high-energy γ-ray events subject to multiple scattering inside the crystal. AMoRE detectors have also demonstrated a fast rise-time of a few ms. This fast rise-time is realized by the use of MMCs and gold phonon collectors and is much faster than that of a crystal detector of similar dimensions equipped with an NTD Ge thermistor, a popular choice in several $0\nu\beta\beta$ experiments. A fast detector is very advantageous in cryogenic $0\nu\beta\beta$ experiments because it reduces unresolved random pileups, which can be one of the most significant background signals in searches for $0\nu\beta\beta$ signals of ¹⁰⁰Mo in particular [163, 235].

Recently, AMoRE finished its pilot stage run, which used 1.9 kg of $^\textrm{48dep}$Ca¹⁰⁰MoO₄ crystals, depleted in ⁴⁸Ca and enriched in ¹⁰⁰Mo, at Yangyang underground laboratory (Y2L). ⁴⁸Ca was depleted because ⁴⁸Ca undergoes double beta decay with a Q-value of 4268 keV, higher than that of ¹⁰⁰Mo, and it can contribute to the background in the ROI that is indistinguishable from the ¹⁰⁰Mo signal. The pilot run resulted in a $T_{1/2}^{0\nu}$ sensitivity of 9.5 × 10²² years [237]. Presently, an upgrade version of AMoRE-I equipped with 6 kg of $^\textrm{48dep}$Ca¹⁰⁰MoO₄ and Li$_2^{100}$MoO₄ just started data collection. Moreover, AMoRE-II has been fully funded, with a plan to build a 200 kg detector of Li$_2^{100}$MoO₄ with a lower cosmic background to achieve a limit sensitivity of $T\,_{1/2}^{0\nu} \gt 5 \times 10^{26}$ years or, equivalently, $m_{\beta\beta} \lt (0.017$–0.029) eV [238]. The experimental site and facility are being prepared for the large-scale superconducting detector of AMoRE-II in Yemilab, a new underground lab in Korea [239].

AMoRE's MMC-based detection scheme has been adopted by another $0\nu\beta\beta$ experiment: CAlcium fluoride for the study of Neutrinos and Dark matters by Low Energy Spectrometer (CANDLES) [240, 241]. The CANDLES experiment has been searching for $0\nu\beta\beta$ signals of ⁴⁸Ca from CaF₂ crystals using liquid scintillators as their main detector. As a strategic R&D project, CANDLES recently carried out R&D experiments in collaboration with AMoRE, successfully demonstrating its PID capability based on simultaneous measurement of heat and light signals from a CaF₂ crystal [240, 241].

Low-temperature thermal calorimetric detectors have long been studied for use in $0\nu\beta\beta$ search experiments [242]. In particular, a series of projects based in Italy have investigated the development of large-mass low-temperature detectors for more than 30 years [243]. Preceded by Milano Double Beta Decay (MiDBD) and CUORICINO, the CUORE project has utilized 800 kg TeO₂ crystals as the source/detector of ¹³⁰Te $0\nu\beta\beta$ events and began collecting data in 2017. All three experiments were conducted in the Laboratori Nazionali del Gran Sasso (LNGS) underground laboratory in Italy [243–245]. A new project called the CUORE Upgrade with Particle Identification (CUPID) is under development as a successor of the CUORE project but with an improved background rejection capability [246].

The CUORE detectors use NTD Ge thermistors as temperature sensors of the TeO₂ crystals. For each crystal, an NTD Ge thermistor is glued onto the crystal surface. In this kind of setup, the heat flow between the crystal and the sensor at low temperatures is governed by phonon transfer between interfaces of difference media and interactions between phonons and electrons in the sensor, as discussed in sections 3.4.1 and 3.4.2. Moreover, heat transfer by athermal processes is greatly suppressed. As a result, the typical rise and decay times of heat signals in the CUORE detectors are relatively slow, on the order of a few hundred milliseconds and several seconds, respectively. This inefficient heat flow was intended to function as an efficient low-pass filter of the detector signals because it can help increase the energy resolution of such a massive detector. Thermistors are a suitable choice for CUORE detectors also because they have a large dynamic range of several MeV. On the other hand, the slow response of the thermistor makes the detector particularly vulnerable to microphonic noise, such as that caused by mechanical vibration, typically with $1/f$ characteristics. Thus, it is critical to minimize the vibration of the cryostat for CUORE. After all these considerations, the CUORE detectors have demonstrated high energy resolution in their ROI (7.7 keV FWHM near 2.5 MeV) [245].

The CUORICINO and CUORE experiments resulted in the best detection limit for $0\nu\beta\beta$ decay of ¹³⁰Te to date (a half-life of 3.2 × 10²⁵ years, which corresponds to the Majorana neutrino mass of $75 \lt m_{\beta\beta} \lt 350$ meV [28]). The next-generation experiment CUPID aims not only to employ scintillating crystals to improve the background through PID but also to investigate other isotopes, such as ⁸²Se and ¹⁰⁰Mo. In particular, the CUPID-Mo experiment based on Li₂MoO₄ crystals with NTD Ge thermistors has been carried out to demonstrate the applicability of CUPID [247]. Moreover, in CUPID-0, the first pilot experiment of CUPID, the PID technique based on heat-light detection resulted in suppression of background signals induced by alpha particles in Zn⁸²Se crystals [248]. Together with AMoRE, CUPID is expected to be a key player in the $0\nu\beta\beta$ searches for the next decade.

The PID capability is critical in $0\nu\beta\beta$ searches to efficiently reject background signals. Although PID using only the heat channel has been demonstrated in a few scintillating crystals [235, 249], in general, simultaneous measurement of heat and light signals using a separate light detector is superior to the single-channel-based PID. A separate light detector may also enable dual-channel-based PID even with a nonscintillating crystal such as TeO₂ [192] because energetic electrons and ions in a dielectric crystal generate a different amount of Cherenkov light. This capability will broaden the selection of crystals and facilitate searching for $0\nu\beta\beta$ in multiple isotopes. Moreover, having a light detector is beneficial not only during $0\nu\beta\beta$ data acquisition but also during detector development and characterization in the R&D stages. This is why many of the current $0\nu\beta\beta$ search experiments (e.g. AMoRE, CUPID, and CUPID-Mo) have adopted scintillating crystals and light detectors. However, the total energy released in the light channel is often very weak. For example, only a few percent of the total energy is carried away by scintillation light from scintillating crystals, or in the case of a nonscintillating crystal, Cherenkov photons with a total energy of 100–200 eV are created by a few MeV electrons absorbed in the nonscintillating crystal [192]. Therefore, the light detector should be carefully designed to maximize the size of the light signals.

The light detector should be implemented with a sensor with high energy resolution and high sensitivity. Superconducting sensors such as TES, MMC, and KID are a good choice [250–252]. These sensors are also advantageous because their intrinsic response time is very fast, and the overall response time can be accelerated by making direct contact between the sensor and the light absorber or by placing a phonon collector film between them, which makes these detectors sensitive to both athermal and thermal phonons. The athermal phonon signal contributes a large portion of the total heat signal in the first few milliseconds and can be utilized to extract extra information that the thermal phonon signal alone cannot reveal.

In addition, if the light signal is still too weak even with a high-sensitivity sensor, NTL phonon amplification can be applied to the light detectors, especially when the absorber is a Si or Ge wafer, as discussed in section 3.6. Large-gain signal amplification has been demonstrated with TES [253] and MMC [194] as well as NTD Ge [192] devices.

The Primakoff effect describes the resonant production of neutral pseudoscalar particles by photons interacting with an atomic nucleus [254]. Its reverse process, the inverse Primakoff effect/scattering, provides an excellent way to detect axions; an axion can decay into two real photons or be converted to a photon through interaction with a virtual photon in an electromagnetic field. The decay rate can be enhanced by a strong magnetic field.

The current most sensitive axion detectors were developed based on the axion haloscope proposed by Sikivie [121]. As shown in figure 21, in an axion haloscope, a microwave cavity is adopted to detect photons produced via the inverse Primakoff effect. The detection sensitivity of the axion-photon conversion is enhanced when the resonant frequency of a microwave cavity matches the axion mass. The resonant frequency of the cavity is adjustable with a tuning rod. An antenna placed in the cavity detects the induced microwave power in the cavity and transfers the signal into the cryogenic receiver chain.

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Because an extremely low conversion rate is expected, several experimental strategies should be used to increase the signal-to-noise ratio (SNR) of photon detection in an axion haloscope. For instance, a microwave cavity can be implemented with a large superconducting magnet to increase the axion-photon conversion rate. Moreover, to minimize the overall noise, the temperature of the microwave cavity should be lowered to less than 100 mK to reduce the thermal noise of the measurement system, and an amplifier with a highest possible gain should be used. A longer measurement time would also increase the SNR and increase the chance of detecting the rare signal.

Because the major noise in an axion haloscope is thermal noise, the SNR can be described by the Dicke radiometer equation [255–257]:

$$\begin{equation} \mathrm{SNR} = \frac{P}{k_\mathrm{B} T_\mathrm{sys}} \sqrt{\frac{t}{b_a}}, \end{equation} \tag{ 20 }$$

where P is the expected signal power, $k_\mathrm{B}$ is the Boltzmann constant, $T_\mathrm{sys}$ is the system noise temperature, t is the integration time, and b_a is the axion signal bandwidth. For given experimental parameters, the signal power P is expressed as:

$$\begin{equation} P \propto g_{a\gamma\gamma}^2 B^2 V Q_\mathrm{L}, \end{equation} \tag{ 21 }$$

where $g_{a\gamma\gamma}$, B, V, and $Q_\mathrm{L}$ are the model-dependent axion-photon coupling constant, magnetic field, cavity volume, and loaded quality factor of the cavity, respectively [256]. The axion signal bandwidth is determined by the energy dispersion of the axion and can be expressed as $b_a = m_{a}v\delta v /c^2 = m_{a}/Q_{a}$, where m_a is the axion mass, v is the axion velocity, $\delta v$ is the velocity dispersion, and Q_a is the axion 'quality factor' on the order of 10⁶ [257].

The number of signal bandwidths that can be simultaneously scanned is $N = Q_{a}/Q_\mathrm{L}$, where $Q_\mathrm{L}$ is assumed to be smaller than Q_a , and the cavity bandwidth b_c is Nb_a . Then, the time $\Delta t$ required to scan a frequency range $\Delta f$ to achieve a given SNR is determined by:

$$\begin{equation} \Delta t = \frac {\Delta f} {b_c} t \propto \Delta f \frac{\mathrm{SNR}^2 T\,_\mathrm{sys}^{2}}{{g_{a\gamma\gamma}}^4 B^4 V\,^2 Q_\mathrm{L} Q_{a} }. \end{equation} \tag{ 22 }$$

This equation shows that the scan rate $r = \Delta f / \Delta t$ for a target SNR is proportional to ${g_{a\gamma\gamma}}^{4}$, or equivalently, the sensitivity of the experiment (the minimum $g_{a\gamma\gamma}$ that can be detected) improves only with $r^{1/4}$, showing the difficulty of the experiment. Thus, it is highly desired also to improve other experimental parameters in equation (22), namely, the system noise temperature, the magnetic field strength, the cavity volume, and the quality factor of the cavity.

The system noise temperature $T_\mathrm{sys}$ is approximately given by the following:

$$\begin{equation} T_\mathrm{sys} = T_\mathrm{phy} + T_{A1} + \frac{T_{A2}}{G_1} + \frac{T_{A3}}{G_1 G_2} + \cdots, \end{equation} \tag{ 23 }$$

where $T_\mathrm{phy}$ is the physical temperature of the cavity and T_Ai and G_i are the noise temperature and the gain of the ith stage amplifier in the receiver chain, respectively [92]. The noise from the amplifiers, particularly that from the front-end amplifier, is a major contributor to $T_\mathrm{sys}$. Thus, for efficient reduction in $T_\mathrm{sys}$, a voltage-tunable microstrip SQUID amplifier (MSA) is often employed as a front-end amplifier, whose resonance frequency can be adjusted between 0.1 and 0.8 GHz with the length of the microstrip and the tuning voltage [258]. For higher-frequency measurements, JPAs can be used. With a modulation frequency of twice the measuring resonance frequency, phase-preserving amplification can be realized in a wide frequency range up to 10 GHz [102].

Currently, most axion haloscope experiments are targeting SNR ∼5. With this SNR, in the case of the Axion Dark Matter eXperiment (ADMX) to be introduced later, if $Q_\mathrm{L}$ and $T_\mathrm{sys}$ are 30 000 and 0.2 K, respectively, it would take about 6 (300) days to scan $\Delta f$ of 100 MHz around 0.74 GHz and achieve 1 KSVZ (DFSZ) level of sensitivity [259]. Thus, in order to reach the DFSZ level sensitivity in a wide frequency range within a reasonable experimental period, the above parameters must be further improved.

The current axion-photon coupling limits of several axion haloscopes are shown in figure 5. Note that most of the limits set by haloscopes are in the 0.1–10 GHz range. This is because the sensitive frequency region of a haloscope is primarily determined by the dimension of the cavity, which cannot be made arbitrarily large or small for practical reasons. For example, a larger cavity is required to lower the probing frequency, but the size of the cavity is limited by the magnet. On the other hand, a small cavity is required to probe the high-frequency area, but this could result in an unmanageably long scan time as expected by equation (22). Below, we introduce some of the major axion haloscope projects and the challenges for the next step.

• RBF and UF: In the early period of experimental axion searches, Rochester–Brookhaven–Fermilab (RBF) [94, 260] and the University of Florida (UF) [95] conducted axion searches using axion haloscopes based on microwave cavities. The RBF experiment probed the frequency range of 1.09–3.93 GHz with a 10 L copper cavity, a 6 T magnet, and a cryogenic amplifier based on a GaAs heterojunction field effect transistor (HFET), also known as a high-electron-mobility transistor (HEMT) with the noise temperature of ∼12 K and the physical temperature of cavity was ∼4.4 K. The UF experiment probed a similar frequency range with a 7 L copper cavity and a 7.5 T magnet. Using cryogenic HFET amplifiers with the noise temperatures as low as 3 K and the operating temperature of ∼2 K, the detection sensitivity was improved by more than 10 times compared to that of the RBF experiment. Although the $g_{a\gamma\gamma}$ limits obtained by these experiments were far above the values predicted by the KSVZ or DFSZ models, they established the foundation of axion search experiments using axion haloscopes.
• ADMX: Based on the successful demonstration by the two axion haloscope experiments of RBF and UF, the ADMX collaboration built an axion haloscope with sub-GHz target frequencies using a large 200 L cavity and a 7.6 T magnet at Lawrence Livermore National Laboratory in 1995, and this device operated until 2010 [90, 261]. Initially, HFETs with a noise temperature slightly above 2 K were used as the first-stage amplifier, but later, MSAs, whose noise temperatures were well below 100 mK, were adopted. From this first-phase experiment, KSVZ axion-photon couplings between 461 and 860 MHz (1.9–3.53 µeV) were excluded at a 90% confidence level.In 2010, ADMX moved to the Center for Experimental Physics and Astrophysics (CENPA) at the University of Washington and achieved several major detector upgrades [91, 93, 259]. One of the major upgrades was a new cooling system. In the previous phase, the ADMX microwave cavity was cooled to approximately 2 K by a pumped liquid helium cryostat. The new setup used a dilution refrigerator (DR), which cooled the cavity and the first-stage amplifier to the 100 mK range. Another major upgrade was the replacement of the first amplifier MSAs with JPAs with a tunable resonance, which enabled searches for axions in the 675–800 MHz (2.79–3.31 µeV) range, extended from the ${\sim}650$ MHz (∼2.69 µeV) region allowed when using the MSAs. The sensitivity of the ADMX with a 140 L cavity in these frequency regions reached the DFSZ level.Recently, the ADMX collaboration has developed a new 'sidecar' cavity to search for axions at higher frequencies (4–7 GHz) that are inaccessible with the main cavity [92]. To increase the resonant frequency, the sidecar cavity has a tiny volume of 0.38 L. The sidecar cavity experiment published new limits in this high-frequency region [92] and is planning to broaden their axion search range with several upgrades in their setup, including the adoption of quantum-limited amplifiers, to be operated in tandem with the main ADMX experiment.
• HAYSTAC: The Haloscope at Yale Sensitive to Axion CDM (HAYSTAC) branched off of ADMX to probe high-frequency axions; HAYSTAC was initially called ADMX-HF. They utilize a 2 L cavity, which is much smaller than ADMX's 140 L cavity. Recently, they reported a near-quantum-limited sensitivity of approximately 5.75 GHz (∼23.8 µeV) using a JPA [262] and a DR. Their first result was near that of the KSVZ prediction except for ADMX [97]. Recently, the detection sensitivity was further improved by using a quantum squeezed-state receiver (SSR), a new technique that circumvents the quantum limit [263]. Using the SSR, the scan rate was doubled, and a detection sensitivity almost reaching the KSVZ prediction was achieved at around 4.14 GHz (∼17.1 µeV).
• CAPP: The Center for Axion and Precision Physics (CAPP) at the Institute for Basic Science (IBS) was established in 2013 to search for axions in a wide frequency range of 1–10 GHz. Recently, CAPP published results from a series of pilot experiments using three different microwave cavities and HEMTs. Lee et al reported the highest sensitivity limit at around 1.62 GHz (∼6.7 µeV) using a 3.5 L cavity, a superconducting 8 T magnet, and a DR [98]. Jeong et al adopted an interesting cavity design called a multiple-cell cavity, a highly efficient way to probe high-frequency regions with a large volume and a relatively simple setup [99]. Using a double-cell cavity, a 9 T magnet and He-3 cryogenic system, they obtained the highest sensitivity limit at around 3.25 GHz (∼13.5 µeV). More recently, Kwon et al reported promising results in the 2.46–2.75 GHz range using a setup composed of two half-cavities, one with a volume of 0.59 L and the other with a volume of 1.12 L, in the same cryostat with an 8 T magnet and a DR [100]. Especially at around 2.59 GHz (∼10.7 µeV), the limit of this setup was just above the KSVZ prediction.There are several ongoing R&D projects within CAPP to enhance the detection sensitivity. These efforts include employing a much stronger magnet, further advancing the multiple-cell cavity approach, and developing a superconducting cavity and quantum-limited amplifiers [264]. For the stronger magnet, they have been testing a new 12 T low-temperature superconducting magnet with a bore size of 32 cm and an 18 T high-temperature superconducting (HTS) magnet with a bore size of 7 cm [265]. Although their 18 T magnet is the strongest magnet ever adopted for axion haloscopes, its relatively small bore limits the size of the cavity. They plan to upgrade the magnet to a 25 T HTS magnet with a 10 cm bore that is under development [266, 267]. For the superconducting cavity, they demonstrated a cavity quality factor six times higher than that of their copper cavity using a polygon-shaped cavity with commercial YBCO tapes covering the entire inner wall [268]. For the quantum-limited amplifiers, R&D on JPAs is ongoing [123]. Employing JPAs as first-stage amplifiers instead of HEMTs will decrease the noise temperature to under 1 K and improve the sensitivity. With all the planned upgrades, CAPP is expected to reach the DFSZ limit in the 0.7–3 GHz range using the 12 T magnet and the KSVZ limit at frequencies up to 10 GHz using the 25 T magnet [264].
• QUAX: QUaerere AXion, or QUest for AXion, (QUAX) is a collaboration searching for axion or ALPs in two different approaches, one based on the axion-photon coupling (the QUAX-$a\gamma$ experiment) and the other on the axion-electron coupling (the QUAX-ae experiment [269]). Here, we focus only on the former.In 2019, they reported the first axion-photon coupling limit around 37 µeV (∼9 GHz) using a 36.43 ml NbTi superconducting cavity with a high $Q_\mathrm{L}$ of 2.0 × 10⁵ and a magnetic field of 2 T [101]. It was the first axion limit result from an axion haloscope built on a superconducting cavity. In 2021, they obtained an improved axion limit close to KSVZ around 43.0 µeV (10.4 GHz) using a 80.56 ml cavity made of Cu with $Q_\mathrm{L}$ of 36 000 and 8.1 T at an operating temperature of 200 mK [102]. The improved limit was also attributed to lowering the noise temperature from 15.3 to 0.9 K using a JPA.QUAX-$a\gamma$ is currently building two detectors, one in the Laboratori Nazionali di Legnaro (LNL) and the other in the Laboratori Nazionali di Frascati (LNF). They are complementary detectors built upon slightly different techniques. The LNL detector will be based on a cavity made of hollow dielectric cylinders covering 10–11 GHz [270] while the LNF detector will be based on a multicavity with the cavities tuned to different frequencies, covering 9–10 GHz [271].

For the search for axions and ALPs in the mass range below 1 µeV (or 0.2 GHz), the use of microwave cavities is not practical and sometimes even impossible because the dimensions of the cavities with such a low resonance frequency will exceed that of any available strong magnets. Instead, a lumped superconducting circuit can be employed to directly sense the electromagnetic signal induced by the photons converted from axions via the inverse Primakoff effect.

Figure 22 describes an axion detection scheme using a resonator circuit. A superconducting pickup loop is placed in a strong magnetic field to boost axion-photon conversion. This scheme overcomes the practical limit of placing a large microwave cavity inside the bore of a fixed-size high-field magnet to search for low-mass (frequency) axions. Note that the maximum magnetic field strength in these experiments is well below the critical magnetic field of the superconducting wire typically made of Nb–Ti alloy (∼15 T). The other end of the pickup loop is coupled to a magnetometer such as a SQUID. Since the coupling constant $g_{a\gamma \gamma}$ predicted by the KSVZ and DFSZ models becomes weaker as the axion mass becomes lighter, the magnetometer should be extremely sensitive for ultralow mass axion searches. The technological maturity of superconducting sensors and superconducting electronics may provide the required sensitivity over a wide frequency region between DC and a few hundred MHz.

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We introduce three projects searching for low-mass axion DM based on superconducting circuits and superconducting sensor technologies.

• ABRACADABRA: A Broadband/Resonant Approach to Cosmic Axion Detection with an Amplifying B-field Ring Apparatus (ABRACADABRA) uses a toroidal magnet that induces a large circular magnetic field and a superconducting pickup loop placed in the center of the toroidal structure. The pickup coil is meant to measure electromagnetic waves along the circular axial volume due to axion-photon conversion under the magnetic field. A sensitive dc-SQUID is inductively coupled to the pickup loop with a superconducting flux transformer to match the inductance of the SQUID input coil and the pickup loop. The detector apparatus also includes a current-injection coil along the axial line of the toroidal coil for calibration purposes. Recently, ABRACADABRA-10 cm, a prototype detector, was built with an active volume of 890 cm³ and a 10 cm 1 T toroidal magnet, and its performance was tested. Broad-band detection was carried out in the m_a range of 0.3–8.1 neV, resulting in a $g_{a\gamma \gamma}$ sensitivity as low as approximately 10⁻¹⁰ GeV⁻¹ [273].
• ADMX SLIC: ADMX Superconducting LC Circuit Investigating Cold Axions (ADMX SLIC) is another project using a superconducting lumped element circuit for axion DM searches. The idea was proposed in [272], and the first measurement in this project was carried out recently [274]. A superconducting LC circuit was introduced into an ADMX-type 7 T magnet, which is a cylindrical magnet with a large bore, as shown in figure 22. A rectangular loop antenna was placed inside the magnet to measure the electromagnetic signals induced by the photons converted from axions. The antenna was made of a copper-matrix-free 0.25 mm diameter NbTi wire strung around a polytetrafluoroethylene structure of 7.62 cm × 31.25 cm. The LC resonance circuit was equipped with a piezoelectric-driven capacitor placed at the low-temperature stage to be able to tune the resonance frequency of the circuit. The first measurement resulted in $g_{a \gamma \gamma}$ sensitivities below 10⁻¹² GeV⁻¹ in the narrow ranges of (1.7498–1.7519) × 10⁻⁷ eV, (1.7734–1.7738) × 10⁻⁷ eV, and (1.8007–1.8015) × 10⁻⁷ eV [274].
• DM Radio: Dark Matter Radio (DM Radio) is another DM detector based on a tunable lumped-element LC resonator [275]. The inductor is a superconducting coil wrapped around a pickup sheath composed of Nb. The capacitor is tunable using movable sapphire dielectrics placed inside a Nb capacitor. The excess power induced by axion-photon conversion above the thermal noise level is sensed by dc-SQUIDs, ac-SQUIDs, or parametric amplifiers depending on the target frequency. The overall detection scheme is similar to that in other projects based on a superconducting LC circuit, but the resonator of DM radio is enclosed in a Nb superconducting shield, which keeps the active detector volume free from electromagnetic noise from the surroundings while having no effect on the axions. This setup allows DM Radio to detect axions as well as ultralight hidden photons (or dark photons). A hidden photon is a hypothetical particle that interacts with normal photons via kinetic mixing with an extremely small mixing angle [276]. Hidden photons have many phenomenological similarities with axions, but unlike axions, their detection does not require a strong magnetic field.A proof-of-principle detector of 0.1 L was designed with an LC circuit with a fixed resonance frequency of 492 kHz and a quality factor of ∼40 000. Since this detector did not have a magnet for axion-photon conversion, it was not sensitive to axions but only to hidden photons. A test run with a total integration time of 5.14 hours resulted in an upper limit on the hidden photon mixing angle of ∼10⁻⁹ at around 2 neV [277]. As a next step, a small-scale (∼0.7 L) detector called DM Radio-Pathfinder was built with a tunable LC resonator with a quality factor of 148 000 [278]. A 1 year scan of the DM Radio-Pathfinder over the 100 kHz to 10 MHz range is expected to set an upper limit on far below 10⁻¹⁰ [277].Based on the success of the pilot projects of ABRACADABRA and DM Radio, the two projects are collaboratively working on a new DM Radio-50L detector. The new detector will be operated under a magnetic field in a DR. After a 1-year scan with a resonator quality factor of 10⁶ and a 0.5 T magnetic field over a wide range between 40 peV and 40 neV (10 kHz–10 MHz), the projected sensitivities for axions and hidden photons are $g_{a \gamma \gamma} \sim 1\times 10^{-15}$ GeV⁻¹ and $\epsilon = 10^{-15}$–10⁻¹², respectively [278]. A future generation detector, DM Radio-M3, with a 1 m³ volume and a 4 T magnet is also proposed. The projected axion sensitivity after a 3.5 year scan may reach the KSVZ prediction above 3 MHz and the DFSZ prediction above 10 MHz [278].